#### Glasnik Matematicki, Vol. 41, No.2 (2006), 217-221.

### ON A FAMILY OF QUARTIC EQUATIONS AND A DIOPHANTINE PROBLEM OF MARTIN GARDNER

### P. G. Walsh

Department of Mathematics, University of Ottawa, 585 King Edward St., Ottawa, Ontario, Canada K1N-6N5

*e-mail:* `gwalsh@mathstat.uottawa.ca`

**Abstract.**
Wilhelm Ljunggren proved many fundamental theorems on
equations of the form *aX*^{2} - *bY*^{4} = δ,
where δ ∈
{±1, 2, ±4}. Recently, these results have been improved using
a number of methods. Remarkably, the equation
*aX*^{2} - *bY*^{4} = -2
remains elusive, as there have been no results in the literature
which are comparable to results proved for the other values of
δ. In this paper we give a sharp estimate for the number of
integer solutions in the particular case that *a* = 1 and *b* is of
a certain form. As a consequence of this result, we give an
elementary solution to a Diophantine problem due to Martin Gardner
which was previously solved by Charles Grinstead using Baker's
theory.

**2000 Mathematics Subject Classification.**
11D25.

**Key words and phrases.** Diophantine equation, integer point, elliptic curve,
Pell equation.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.41.2.04

**References:**

- J. H. Chen and P. Voutier,
*Complete solution of the Diophantine equation **x*^{2}+1=d*y*^{4}
and a related family of quartic Thue equations, J. Number Theory
**62** (1997), 71-99.

MathSciNet
CrossRef

- M. Gardner,
*On the patterns and the unusual properties of figurate
numbers*, Sci. Amer. **231** (1974), 116-121.

- C. M. Grinstead,
*On a method of solving a class of Diophantine equations*,
Math. Comp. **32** (1978), 936-940.

MathSciNet
CrossRef

- D. H. Lehmer,
*An extended theory of Lucas' functions*, Ann. of Math. (2)
**31** (1930), 419-448.

MathSciNet
CrossRef

- W. Ljunggren,
*Zur Theorie der Gleichung **x*^{2}+1=D*y*^{4},
Avh. Norske Vid. Akad. Oslo (1942), 1-27.

MathSciNet

- W. Ljunggren,
*Ein Satz über die Diophantische Gleichung
**Ax*^{2}-*By*^{4}=*C* (*C*=1,2,4),
in: Tolfte Skandinaviska Matematikerkongressen,
Lund, 1953, Lunds Universitets Matematiska Inst., Lund, 1954,
188-194.

MathSciNet

- W. Ljunggren,
*On the diophantine equation **Ax*^{4}-*By*^{2}=*C*
(*C*=1,4), Math.
Scand. **21** (1967), 149-158.

MathSciNet

- W. Ljunggren,
Collected Papers of Wilhelm Ljunggren, edited by
Paulo Ribenboim,
Queen's University, Kingston, 2003.

MathSciNet

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